A four digit number (numbered from $0000$ to $9999$) is said to be lucky if the sum of its first two digits is equal to the sum of its last two digits. If a four digit number is picked up at random, then the probability that it is lucky number is 

The total number of ways of choosing the ticket is $10000.$ 

Let the four digits number on the ticket be $x_1{x}_2{x}_3{x}_4,$  Note 

that $0\le x_1+x_2\le 18$ and $0\le x_3+x_4\le 18$.

Also, the number of non-negative integral solutions of $x+y=m$

(with $0\le x,y\le 9$) is $\left\{\begin{array}{l}m+1 if 0⩽m\le 9\\ 19-m if10\le m\le 18\end{array}\right.$ 

                             $\begin{array}{l}\mathrm{The} \mathrm{number} \mathrm{of} \mathrm{favourable} \mathrm{ways}=1\times 1+2\times 2+\dots +10\times 10+9\times 9+8\times 8+\dots \\ +1\times 1\\ =2\left\{\frac{9\times 10\times 19}{6}\right\}+100=670\end{array}$n(E) = 670

$\therefore$ probability of required event $=\frac{670}{1000}=0.067.$